A Predictive Roadmap for Stochastic Scaling Exponents Across Universality Classes: Identifying the Mathematical Gaps

This memorandum serves as a predictive roadmap for the application of the universal bivariate scaling law, Vₑff (μ, σ) = μ^γ * F (σ / μ^κ), across divergent topological routes to chaos. By isolating the universal scaling architecture from class-specific geometric constants, we identify the solved eigenvalues for established universality classes and explicitly define the unmapped stochastic eigenvalues required to resolve higher-order and non-unimodal critical transitions. A validation taxonomy is provided to delineate confirmed empirical baselines from theoretical predictions. Keywords: Universality Classes Renormalization Group Theory Scaling Exponents Stochastic Perturbation Critical Transitions Bifurcation Theory Nonlinear Dynamics